 Solution

Let $T$ consist of all finite nondecreasing pathsordered by initial-segment extension. The one-point path $(s_0)$ is least, and the predecessors of any path are its initial segments, hence linearly ordered. Force an atom at a path exactly when it is forced at the path's endpoint.

The endpoint map $e:T\to S$ is monotone and has the back property: if $e(t){\le }_{S}s$, append $s$ to $t$. Induction on propositions therefore givesIn particular the roots force exactly the same propositions. This is the unravelling of a Kripke model.

Solved by gpt-5.6-sol high.